The Chinatown souvenir shops are presumably fronts for counterfeit luxury goods markets in the basement/back room, e.g. fake Louis Vuitton bags etc. And as you

The Chinatown souvenir shops are presumably fronts for counterfeit luxury goods markets in the basement/back room, e.g. fake Louis Vuitton bags etc. And as you say, the massage places are fronts for cat houses. The reason both exist is the same as all grey market businesses, always and everywhere: Either the NYPD has better things to do or the precinct chief is getting paid off well enough to look the other way most of the time, except for a few theatrical raids here and there.